A Comprehensive Experimental Database for the Calibration ... · t(E= Young’s modulus; f0= tensile strength) which was de-rived based on Irwin’s approximation for the size of

Noname manuscript No.(will be inserted by the editor)

Characterization of Concrete Failure Behavior

A Comprehensive Experimental Database for the Calibration

and Validation of Concrete Models

Roman Wendner · Jan Vorel · Jovanca

Smith · Christian Hoover · Zdenek P.

Bazant · Gianluca Cusatis*

Received: date / Accepted: date

Abstract Concrete is undoubtedly the most important and widely used con-

struction material of the late 20th century. Yet, mathematical models that

can accurately capture the particular material behavior under all loading

conditions of significance are scarce at best. Although concepts and suitable

models have existed for quite a while, their practical significance is low due

to the limited attention to calibration and validation requirements and the

scarcity of robust, transparent and comprehensive methods to perform such

tasks. In addition, issues such as computational cost, difficulties associated

Roman WendnerChristian Doppler Laboratory, University of Natural Resources and Life Sciences Vienna,Peter-Jordanstr. 82, 1190 Vienna, Austria.

Jan VorelCzech Technical University in Prague, Thakurova 7, 16629 Prague, Czech Republic.University of Natural Resources and Life Sciences Vienna, Peter-Jordanstr. 82, 1190 Vienna,Austria.

Christian G. HooverMassachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139,USA.

Jovanca Smith, Zdenek P. Bazant, Gianluca Cusatis*Northwestern University, Technological Institute, 2145 Sheridan Road, Evanston, IL 60208,USA.

*corresponding author: Tel.: +1 (847) 491-4027E-mail: [email protected]

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2 Roman Wendner et al.

with calculating the response of highly nonlinear systems, and, most impor-

tantly, lack of comprehensive experimental data sets have hampered progress

in this area. This paper attempts to promote the use of advanced concrete

models by (a) providing an overview of required tests and data prepara-

tion techniques; and (b) making a comprehensive set of concrete test data,

cast from the same batch, available for model development, calibration, and

validation. Data included in the database ’http://www.baunat.boku.ac.at/cd-

labor/downloads/versuchsdaten’ comprise flexure tests of four sizes, direct ten-

sion tests, confined and unconfined compression tests, Brazilian splitting tests

of five sizes, and loading and unloading data. For all specimen sets the nominal

stress-strain curves and crack patterns are provided.

Keywords 3-point bending · Brazilian splitting · size effect · cohesive

fracture · softening · single notch tension

1 Introduction

Rapid progress in concrete technology in recent years has led to the develop-

ment of many new construction materials with novel properties. These are,

among others, ultra-high performance concretes (UHPC) with strengths of up

to 200 MPa [1], self consolidating concretes (SCC) with improved rheology [2],

fiber reinforced concretes (FRC) characterized by significantly increased duc-

tility [3,4], and engineered cementitious composites (ECC) with superior im-

pact resistance [5].

This development provides many opportunities for the construction indus-

try, associated with as many challenges. The main obstruction for a wide-

spread application of these novel materials is a significant lack of experience.

Traditionally, design codes were developed based on large experimental in-

vestigations and multi-decade practical experience. Thus, safe design rules

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Characterization of Concrete Failure Behavior 3

could be ensured that are typically characterized by sufficiently conservative

assumptions. This approach, however, can no longer satisfy the requirements

of modern construction industry and it is increasingly less sustainable from

the economic point of view. The only feasible solution is supplementing exper-

iments with analytical predictions based on accurate, reliable, and validated

models. Simulations can provide the means for virtual testing of structural

capacity, performance, and, ultimately, also life-time, if structural analysis is

coupled with multi-physics and deterioration modeling.

The main characteristics of the tensile behavior of concrete and other quasi-

brittle materials are cracking and strain softening – a loss of carrying capacity

for increasing deformation. Such behavior is typically described by non-linear

fracture mechanics and suitable strain softening laws, characterized by the to-

tal fracture energy GF or, equivalently, by Hillerborg’s characteristic length [6,

7], lch = EGF /f′2t (E= Young’s modulus; f ′t = tensile strength) which was de-

rived based on Irwin’s approximation for the size of the plastic zone in ductile

materials [8,9]. The most important effect of strain softening is the depen-

dence of structural strength on structural size [8] – any mathematical model

for concrete that has any value must be able to simulate size-effect. Concrete

behavior in compression is even more complicated: under low or no confine-

ment the compressive behavior features brittleness and strain-softening; for in-

creasing confinement, however, the behavior transitions from strain-softening

to strain-hardening and it is characterized by significant ductility. Under suf-

ficient lateral confinement concrete can reach strains over 100% without the

loss of load carrying capacity and visible damage [10–14].

Over the years many constitutive models have been developed to describe

the behavior of concrete. They utilize the concepts of plasticity [15,16], damage

mechanics [17,18] or fracture mechanics [8,18] and they are typically formu-

lated in tensorial form by using the classical framework of continuum me-

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chanics. Continuum constitutive equations can also be formulated in vectorial

form through the microplane theory [19,20], which has a number of advan-

tages over tensorial formulations. Microplane models do not need to be formu-

lated as functions of macroscopic stress and strain tensor invariants [21] and

the principle of frame indifference, however, is satisfied by using microplanes

that sample (without bias) all possible orientations in the three-dimensional

space. The constitutive laws specified on the microplanes are activated by

employing either the kinematic or the static constraints. The former defines

the microplane strains as projections of the macroscopic strain tensor whereas

the latter defines the microplane stresses as projections of the macroscopic

stress tensor. Kinematically constrained formulations can be used with mi-

croplane constitutive laws exhibiting softening and for this reasons they have

been adopted for quasi-brittle materials [22] such as concrete [23–25], even at

early age [26], rocks [21], and composite laminates [27,28].

For continuum formulations, objectivity of the solution and independence

of the numerical solution upon the finite element discretization have to be

either inherent to the constitutive model, as for example in the case of high

order [29] and nonlocal [30,31] models, or must be imposed using regularization

techniques such as the crack band approach [32,8]. Methods that do not suffer

from mesh sensitivity are also the ones accounting for strain softening through

the insertion of cohesive discrete cracks [6,33–36].

Another class of models often used to simulate quasi-brittle materials is

based on lattice or particle formulations in which materials are discretized “a

priori” according to an idealization of their internal structure. Particle size

and size of the contact area among particles, for particle models, as well as

lattice spacing and cross sectional area, for lattice models, equip these type of

formulations with inherent characteristic lengths and they have the intrinsic

ability of simulating the geometrical features of material internal structure.

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This allows the accurate simulation of damage initiation and crack propagation

at various length scales at the cost, however, of increased computational costs.

Earlier attempts to formulate particle and lattice models for fracture are

reported in [37–43] while the most recent developments were published in a

Cement Concrete Composites special issue [44]. A comprehensive discrete for-

mulation for concrete was recently proposed by Cusatis and coworkers [45–50]

who formulated the so-called Lattice Discrete Particle Model (LDPM). LDPM

was calibrated, and validated against a large variety of loading conditions in

both quasi-static and dynamic loading conditions and it was demonstrated to

possess superior predictive capability.

As evident from the short review presented above, many different concrete

material models are available in the literature. However, the challenge that still

remains is selecting the model that is most suitable for a given application, and

obtaining the required input parameters. These can either have direct physical

meaning or be solely model parameters that have to be inversely identified.

However, in all cases sufficient experimental data are required to uniquely de-

termine and finally validate the model parameters. This entails the availability

of data of all required test types with sufficient number of samples to yield

statistically meaningful results. Required tests include, but are not limited to,

uniaxial compression, confined compression or triaxial tests, and direct or in-

direct tension tests. From these strength and modulus information as well as

hardening parameters can be derived. Due to the brittle nature of concrete in-

direct tension tests such as 3-point-bending or splitting are generally preferred.

In order to ensure unique softening parameters, softening post-peak data for

at least two sizes [51] or alternatively two different types of tests, should be

obtained. For fiber reinforced concrete (FRC) bond properties and fiber char-

acteristics are to be determined additionally [52]. Further tests are required

if predictions under high loading rates, or extreme environmental conditions

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have to be carried out. While for established models the predictive capabilities

can be assumed to be satisfactory after calibration, new models also need to

be validated. This step includes a subdivision of tests and specimens into sub-

populations for calibration and prediction, where a random subset (typically

1/2 to 2/3) is allocated to calibration and the rest (ideally encompassing tests

of all types) are used for prediction and validation.

This paper focuses on the discussion of relevant tests for the calibra-

tion and validation of concrete models. In particular, a comprehensive set

of tests including uniaxial compression, confined compression and size-effect

tests in 3-point bending and splitting is presented. All specimen were cast

from the same batch and most were tested at an age of around 400 days with

the exception of standard 28-day compressive strength tests and a few ad-

ditional tests that were carried out at 950 days. This practice ensured that

the change of material properties during the period of testing was negligible.

The raw data obtained in more than 257 tests was post-processed to pro-

vide statistical indicators for material properties and mean response curves

for all types of tests including post-peak softening. The complete collection of

pre-processed response curves as well as the raw data are freely available at

http://www.baunat.boku.ac.at/comprtest.html.

2 Scope of Experimental Investigation

The scientific literature contains an abundance of experimental data covering

different phenomena and mechanisms. However, the number of publications

reporting response curves for uniaxial compression, confined compression, di-

rect and indirect tension of the same concrete are very limited, and basically

none simultaneously provide post-peak response for various sizes.

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The present investigation represents an extension of a size-effect investiga-

tion in 3-point bending conducted by Hoover et al. [53] during which a total of

164 concrete specimens were cast in one batch (see section 2.1) in early 2011

and tested in 2012. A similar investigation dedicated to the fracture properties

of self-consolidating concretes of various compositions is presented in Beygi

et al. [54]. Afterwards, additional 105 specimens were cut from the remain-

ing shards in order to supplement, among others, confined compression tests,

Brazilian splitting tests, direct tension tests, and hysteretic loading-unloading

tests. For all fracture tests of the initial and extended investigation the crack

pattern was documented photographically and digitized by photogrammetric

means. In detail, response curves for the following tests are available:

– 128 three-point bending tests of 400 day old geometrically scaled un-

reinforced concrete beams of four sizes with a size range of 1:12.5 in-

cluding un-notched specimens and beams with relative notch depths of

α = a/D =0.30, 0.15, 0.075, 0.025, see Fig. 1(a).

– 12 centrically and eccentrically loaded 466 day old 3-point bending speci-

mens of depth size D = 93 mm according to Fig. 1(a), with and without

unloading cycles in the softening regime.

– 40 Brazilian splitting tests, of roughly 475 day old prismatic specimens of

5 sizes with a size range of 1:16.7, see Fig. 1(b).

– 12 standard ASTM modulus of rupture tests [55] at 31 days and 400 days,

see Fig. 1(c).

– 24 uniaxial compression tests of 75x150 mm (3”x6”) cylinders at 31 days

and 400 days, see Fig. 1(f).

– 22 uniaxial compression tests of approximately 470 day old cubes of side

lengths D = 40 mm and D = 150 mm, loaded partly monotonically and

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partly with several loading-unloading cycles in the softening regime, see

Fig. 1(d).

– 6 uniaxial compression tests of approximately 950 day old cubes of side

length D = 40 mm

– 6 uniaxial tension tests of approximately 950 day old prisms, see Fig. 1(g)

– 4 confined compression tests of 560 day old cored cylinders of diameter D =

50 mm and length L = 40 mm including 4 unconfined uniaxial compression

tests of cored companion specimens.

– 11 torsion tests of prisms with constant thickness W = 40 mm and depths

D = 40, 60, 80 mm according to Fig. 1(e).

Full or partial post-peak data are available for all tests except the ASTM

modulus of rupture tests and the confined compression tests. Due to the mul-

titude of sizes and specimen geometries all plots are presented in terms of

nominal stress σN and nominal strain εN , the definition of which is given in

the respective subsections of section 3.

2.1 Mix properties and curing

All 164 specimens of the initial investigation (128 beams, 12 ASTM beams,

24 cylinders) were cast in one batch of ready-mixed concrete with a specified

compressive strength f ′c = 31 MPa. On top of 354.8 kg/m3 cement (CEM I,

ordinary portland cement) the mix design includes 59.9 kg/m3 fly ash, result-

ing in a water-cement ratio w/c = 0.41, and a water-binder ratio w/b = 0.35,

respectively. The aggregate to binder ratio a/b = 4.43 is achieved through the

addition of 863 kg/m3 sand and 1015 kg/m3 coarse aggregate (pea gravel, con-

sisting of glacial outwash deposits from McHenry, Illinois) with a maximum

aggregate diameter of 10 mm. The mix further contains 0.78 kg/m3 of water

reducer, 2.08 kg/m3 superplasticizer and 0.56 kg/m3 retarder. The addition

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Fig. 1 Specimen geometry: (a) three point bending tests, geometrically scaled in four sizes,(b) Brazilian splitting tests, geometrically scaled in five sizes, (c) ASTM modulus of rupturetests [55], (d) unconfined compression of cubes of two sizes, (e) torsion tests, (f) unconfinedcylinder tests of two sizes, and (g) single notched tension tests

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of these admixtures was required to guarantee a consistent 150 mm of slump

throughout the 3 hour casting.

The 128 beams of the original size effect investigation were cast horizontally

in molds with constant depth of W = 40 mm. Immediately after casting the

beams and cylinders were covered with plastic and remained untouched for

36 hours after which they were demolded and relocated to a fog-curing room.

There they were stored under ambient temperature conditions (around 23◦C)

and at roughly 98% relative humidity until testing. Immediately after testing

the shards of the bending test specimens were again stored in the fog-curing

room. Special attention was placed on minimizing the transfer time between

testing and storage to avoid uncontrolled shrinkage cracking.

The additional investigations were performed on specimens that were cut

or cored out of the shards of the original investigation. Special attention was

placed on avoiding areas of high stress concentration (support points) and pre-

damaged areas including e.g. the original fracture process zone (FPZ). The

additional specimens as well as the notches were cut with a diamond coated

band saw with water cooling. In each case the exposure time to unsaturated

humidity conditions was limited to a minimum.

2.2 Overview of material properties

The basic concrete properties have been extracted from various tests per-

formed at different ages. For convenience a summary is provided in Table 1,

see also Hoover et al. [53]. In addition to the experimental results the inversely

obtained 400-day modulus, extracted from 93-mm 3-point bending tests, is pre-

sented. Relevant fracture parameters are given in Table 3. Where possible the

inherent scatter of the experiments is quantified by the coefficient of variation

CoV = std/mean. The high carefulness in casting, specimen preparation, and

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Table 1 Material properties extracted from cylinder tests and ASTM modulus of rupturetests

material property unit mean CoV [%]

compressive cylinder strength fcyl,75(31) MPa 46.5 3.2compressive cylinder strength fcyl,75(400) MPa 55.6 3.7compressive cube strength fcu,40(470) MPa 56.6 9.5compressive cube strength fcu,150(470) MPa 57.1 5.5compressive cube strength fcu,40(950) MPa 61.2 8.2

ASTM modulus of rupture fr(31) MPa 6.7 5.2ASTM modulus of rupture fr(400) MPa 8.3 3.6

modulus of elasticity, 75 mm cyl Ecyl,75(31) GPa 27.74 6.2modulus of elasticity, 75 mm cyl Ecyl,75(400) GPa 34.38 3.9modulus of elasticity, D=40mm Er,40(400) GPa 35.70 7.0modulus of elasticity, D=93mm Er,93(400) GPa 41.29 6.8modulus of elasticity, D=215mm Er,215(400) GPa 43.68 9.4modulus of elasticity, D=500mm Er,500(400) GPa 43.66 12.7modulus of elasticity, inverse Einv(400) GPa 37.94poisson ratio ν - 0.172 10.0

testing was rewarded by a remarkably low experimental scatter with coefficient

variation less than 10% and a minimum of statistical outliers. Only two of the

early age compression cylinders and one of the beams failed the Grubb’s test

for outliers [56,57], assuming a significance level of α = 0.05. The respective

specimens were excluded from all further analyses. Compressive strength was

determined based on 75x150 mm cylinders, 40 mm and 150 mm cubes. The re-

ported Poisson’s ratio was determined based on the circumferential expansion

of standard cylinders in compression [53].

The strength and modulus development can be predicted by Eq. (1) accord-

ing to the fib Model code 2010 with parameter s = 0.25 for R-type cement

[58] and a reference age tref = 28 days.

f(t) = f28 βfib(t), E(t) = E28

√βfib(t), βfib(t) = es(1−

√tref/t) (1)

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The equivalent ACI formulation is given by Eq. (2) with a = 4.0 and b =

0.85 for Type-I cement [59].

f(t) = f28 βACI(t), E(t) = E28

√βACI(t), βACI(t) =

t

a+ bt(2)

While the model code prediction for the strength development is very good

and the ACI prediction is fair, both formulations fail to predict the modulus

development. In Table 2 the extracted 28-day strength and modulus values

are given including the 95% confidence bounds and the root mean square

error of the fit. In order to fit the modulus development data the square-root

dependency on strength would have to be replaced by the fitted exponents

5/3 for the ACI model (ACI*,[59]), and 5/4 respectively for the fib model

(fib*,[58]).

The Young’s modulus can be predicted from compressive strength utiliz-

ing either the ACI (Eq. (3)) or the fib (Eq. (4)) formulation. The parame-

ters E0 and αE in the latter are dependent on aggregate type. The default

values for quarzite are E0 = 21.50 GPa and αE = 1.0. The ACI predic-

tion for the 28-day Young’s Modulus yields 32.39 GPa and overestimates the

experimentally identified value of 29.81 GPa. The fib equation gives an ini-

tial tangent stiffness of 35.78 GPa which corresponds to a reduced modulus

Ec = (0.8 + 0.2fcm/88MPa)Eci = 32.38 GPa, a value almost identical with

the ACI prediction.

E28,ACI = 4734√f28 (3)

E28,ci,fib = E0 αE (f28/10MPa)1/3 (4)

Hoover et al. [60] studied the size-effect [8] in 3-point-bending based on the

presented data-set. The values he obtained for the initial fracture energy Gf

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Table 2 Strength and modulus development: quality of fit expressed in terms of root meansquare error (RMSE; ∗ stands for the models with fitted exponents)

parameter model value unit RMSEfcyl,75(28) fib 46.1 [45.6, 46.6] MPa 0.184fcyl,75(28) ACI 46.8 [43.4, 50.3] MPa 1.243fr(28) fib 6.8 [6.4, 7.2] MPa 0.158fr(28) ACI 6.9 [6.0, 7.8] MPa 0.314Ecyl,75(28) fib 29.63 [23.88, 35.37] GPa 1.988Ecyl,75(28) ACI 29.81 [23.08, 36.54] GPa 2.313Ecyl,75(28) fib* 27.31 GPa —Ecyl,75(28) ACI* 26.78 GPa —

Table 3 Fracture parameters according to Hoover et al. [60]

material property unit mean CoV [%]

fitting of Type 2 SEL, α = 0.30initial fracture energy Gf N/m 51.87 —characteristic length cf m 23.88 —

fitting of Type 2 SEL, α = 0.15initial fracture energy Gf N/m 49.78 —characteristic length cf m 20.99 —

work of fracture method, α = 0.30total fracture energy GF N/m 96.94 16.9

work of fracture method, α = 0.15total fracture energy GF N/m 111.1 20.7

by inverse fitting of Bazant’s Size Effect Law [8] as well as for the total fracture

energy GF based on the work of fracture method with bilinear softening stress-

separation curve [7,61,62] are given in Table 3. As expected, the initial fracture

energy accounts for approximately 50% of the total fracture energy. The fib

prediction of GF = 73f0.18cm = 150.5 J/m2 (with fcm in MPa) overestimates

the experimentally obtained value by 50%. Further comparisons with empirical

equations can be found in [60].

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3 Detailed Description of Tests

3.1 Test setup, instrumentation and control

All the tests were performed on servo-hydraulic closed-loop load frames with

capacities of 4.5 MN, 980 kN, and 89 kN respectively. Uniaxial compression

and confined compression tests were carried out in the 4.5 MN load frame.

For the ASTM standard modulus of rupture tests [55], the notched and un-

notched 3-point bending tests, as well as splitting tests of specimens with size

D = 215 mm and D = 500 mm the 1000 kN load frame was used. All the

remaining specimens were tested in the 90 kN load frame. In general a loading

time of 5 minutes to peak, based on preliminary predictions, was attempted.

During specimen preparation all relevant dimensions were rigorously recorded.

The initial condition of each specimens as well as the crack pattern of the

failed specimens were documented with pictures, the latter of which served for

the crack path digitization reported in section 3.6. Piston movement (stroke),

force, and loading time are available with a sampling frequency of 1 Hz for

all specimens. Additionally, the test specific quantities load-point displace-

ment, circumferential expansion, axial shortening, and crack mouth opening

displacement (CMOD) were recorded. Further information can be found in the

test specific sections 3.4 to 3.8.

3.2 Test control and stability

A particular challenge in the testing of quasi-brittle materials is associated with

obtaining post-peak softening response for different types of tests and, ideally,

various sizes. This information is quintessential for calibrating the model pa-

rameters which control softening in tension, shear-tension, or compression un-

der low confinement. The ability to control a specimen in the softening regime

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is a stability problem, which, in its most general form, can be described by an

energetic stability condition formulated in terms of the second variation of the

potential energy δ2Π > 0 [17]. It can be shown that under displacement con-

trol the equilibrium path remains stable as long as no point with vertical slope

is reached. This limit state is called “snap-down” and represents the transi-

tion to a “snap-back” instability which is characterized by an equilibrium path

with global energy release. In the softening regime parts of the system (load

frame as well as specimen) are elastically unloading, releasing energy. Depend-

ing on the experiment the contributions of both vary and in many cases they

are negligible. While the energy release within the specimen is out of the con-

trol of the experimenter (and can also be observed in numerical simulations),

the energy released by the unloading load frame can be controlled through its

compliance.

The practical problem of a specimen with elastic stiffness Kel in a load-

frame with stiffness Km is equivalent to a serial system with total current tan-

gential stiffness K(u) = {1/Km + 1/ [Kel −∆K(u)]}−1, where [Kel −∆K(u)]

= true tangential specimen stiffness, and ∆K(u) = total stiffness change due

to softening. A stable test in displacement control in the softening regime is

possible if ∆u > 0, equivalent to 1/K(u) 6= 0 along the entire equilibrium

path. Consequently, a minimum machine stiffness is required for stable tests,

given by the condition Km > Kel −∆K(u).

In Fig. 2(a) the problem of stable displacement control is illustrated utiliz-

ing experimental data for a 93 mm beam with a relative notch depth α = 0.15

according to Fig.1(a). The observed mean load-displacement curve (thick solid

line), plotted in terms of nominal strain εN and stress σN , see section 3.6,

clearly exhibits a significant snapback. Stable control in neither force nor dis-

placement control (stroke) is possible. However, any observed response from a

stably controlled test (e.g. using CMOD control, see below) can be corrected to

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strain εN [10−3]stress

σN

[MPa]

L =223.9 mm ±0.0%D =93.3 mm ±0.5%W =40.9 mm ±1.0%Ec =36336.7 MPa ±9.5%fr =4.5 MPa ±7.4%number of specimen: 8

0 1 2 30

1

2

3

4

5

Fig. 2 Stability of fracture tests: (a) observed, inelastic, corrected, and critical response for93 mm beams and α = 0.15, (b) observed stable load vs. opening curves for 93 mm beamsand α = 0.15

account for the machine compliance and to recover the proper elastic stiffness

Kel. The results are the inelastic strain component εinel (thin dotted line) and

the unbiased specimen response εspec (thin solid line).

εinel = εobs − σ (1/Kobs) (5)

εspec = εobs + σ (1/Kel − 1/Kobs) (6)

In the example discussed above the specimen itself exhibits no snapback

and an entirely stable test in displacement control would have been possible

for a machine stiffness Km > Kcrit = 5.9 GPa, where Kcrit is the slope of

the steepest section of the softening response. The critical state of softer load

frames with a stiffness Kmi less than Kcrit may be found by drawing a tangent

of downward slopeKmi as sketched in Fig. 2(a) forKmi = 3 GPa. Approximate

machine compliances for the used load frames are given in Hoover et al. [53].

Practically speaking, a stable post-peak test is possible if a monotonously

increasing continuous quantity for control can be found. Up to this point, dis-

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


placement control based on piston movement or load-point displacement was

assumed. However, many more displacement measures exist which exhibit the

desired properties. True crack mouth opening displacement (CMOD) control in

fracture tests or circumferential expansion control in unconfined compression

tests are unconditionally stable. Crack initiation specimens such as un-notched

beams or splitting prisms can be controlled by average strain, or, in general,

by relative displacements between given points on a specimen that include the

forming macro-crack. In the latter case the appropriate gauge length is deter-

mined by two contradictory requirements. The amount of elastically unloading

material within the monitored distance has to be minimized while the gauge

length must be large enough to contain the forming crack with high likelihood.

The success of fracture tests ultimately depends on the proper selection

of specimen geometry, test setup, sensor instrumentation, and control mode.

Thus, during the development of experimental campaigns compliance tests

of the load frame, fixtures, and preliminary simulations of the specimens to

determine a stable mode of control are highly advised.

3.3 Data preparation and analysis

All presented data are processed automatically, thus ensuring unbiased and ob-

jective results. The actual pre-processing is limited to determining the statis-

tics of the specimen dimensions, the automatic removal of pre-test and post-

test data and the application of a low-pass butterworth filter of order 12 with

a normalized cut-off frequency of 0.10 to remove random high frequency noise.

Initial setting in load displacement diagrams is removed by linear extrapo-

lation of the fully elastic part of the loading branch and subsequent shifting by

the respective displacement intercept. The linear region is assumed to occur

in the range (0.50− 0.90)σpeak for beams, (0.25− 0.90)σpeak for compression

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


specimens, and (0.60 − 0.90)σpeak for splitting specimens. For the latter this

range is particularly small due to the high compliance of the used wooden

support strips.

For convenience and readability of the figures the results of the individual

specimen families are reported in terms of a mean response curve and envelope

only. The full set of test curves is available electronically. The mean response

curve is obtained by separately averaging the pre-peak and post-peak branches

of the normalized curves. Each individual curve is scaled by the inverse of

its peak stress σpeak and peak strain εpeak such that all peaks coincide with

coordinates (1, 1). The normalized mean response curve is transformed back

to match the mean nominal stress and mean nominal strain of the group

of curves. Due to the possibility of snap-back strains are averaged at given

stress. Fig. 3 shows the normalized individual load-CMOD curves and the

obtained normalized mean response curve for a 3-point bending test with depth

D = 93mm and non–dimensional notch depth α = 0.15. The small variability

in elastic stiffness and high similarity of the curves in the post-peak regime is

illustrated.

The stroke measurements, which are the basis for some of the load displace-

ment diagrams, obviously include contributions from the compliant load frame

and fixtures. Only on-specimen deformation measurements are not distorted

and can serve for the determination of the true elastic modulus. For model cal-

ibration and validation the elastic response obtained from global deformation

measurements has to be corrected. Assuming that the recorded response can

be decomposed in superposed inelastic and linear elastic strain contributions,

Eq. (6) is applicable for the data correction. Note that, in order to preserve

the scatter in the elastic regime, Kobs is set equal to the elastic loading slope

of the mean response for all specimens of the family.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


normalized nominal strain εN/εpeak

norm

alizednominalstress

σN/σpeak

0 1 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1mm

ab

r

Fig. 3 3-point bending: (a) normalized load-CMOD curves and average response curve bythe example of 3-point bending tests with D = 93mm and α = 0.15, (b) microscopic imageof cut notch tip

3.4 Unconfined compression

The most traditional tests to characterize concrete are unconfined compression

tests, typically performed on cylinders or cubes. The Eurocode [63,64] allows

testing both cubes of 150 mm side length and cylinders with 150 mm diameter

and 300 mm height. The derived cylinder strength fcyl and cube strength fcu

are used to specify concrete with the typical label ’Cfcyl/fcu’. Historically,

200 mm cubes have been used in some countries which might be of signif-

icance for the analysis of historic buildings. The standard compression tests

according to ASTM C39 [65] are performed on 150x300mm (6”x12”) cylinders.

Unconfined compression tests not only yield a material’s uniaxial compressive

strength but also provide insight into the damage behavior already starting

before the peak-load.

In the present investigation twelve 75x150 mm cylinders each were tested

after 31 days and after 400 days, in the middle of the 3-point-bending size-

effect investigation. Additionally, eight 150 mm cubes were cut out of the

undamaged parts of the ASTM modulus of rupture specimens and fourteen

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


40 mm cubes were cut from the remainders of the 3-point-bending size effect

investigation. The tests were performed approximately at an age of 470 days

in parallel with the Brazilian splitting size effect investigation. During loading

only the top load platen was allowed to rotate, in agreement with the ASTM

testing standard [65]. All cylinders and the 40 mm cubes were capped with

a sulfur compound to ensure initially co-planar and smooth loading surfaces.

The 150 mm cubes were loaded on two of the uncut faces that had been

cast against the mold. The observed fracture patterns conformed to Type 1

according to ASTM C39 [65] with reasonably well formed cones on both ends.

Similarly, pyramid-shaped cones were observed in the cube tests.

It is important to note that friction between specimen and loading platens

significantly influences the obtained peak loads by introducing some degree

of lateral confinement in the contact surfaces. While this situation can be

mimicked in numerical analyses by suitable boundary conditions, standard

analyses assuming ideal uniaxial conditions are thwarted. Typically, this effect

is mitigated through the use of friction reducing coatings on the steel platens,

teflon sheets, or elastomeric pads. Unfortunately, this practice is limited to

normal strength concretes as the compressive strength of modern ultra high

performance (UHPC) concretes exceeds the strength of typical friction reduc-

ing materials. In case of this investigation solely sulfur compound capping was

applied.

The recorded specimen dimensions for cubes comprise the length of all

edges as well as the height before and after capping. During testing in addition

to force F (recorded by a 450 kN load cell) and machine stroke δ the platen to

platen distance u was measured by four equiangularly distributed LVDTs of

±2.5mm travel, see Fig. 1(d). The chosen stable mode of control was machine

stroke.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


In the case of the cylindrical specimens two perpendicular diameters D

were recorded for each end in addition to four height measurements L each

before and after capping. Similar to the cube tests force F and piston stroke

δ were recorded. The test was controlled by circumferential expansion ∆p

after a pre-load of roughly 50% of peak was applied. The axial shortening

of the specimen was measured by four equiangularly distributed LVDTs held

in place by two symmetric rings with a nominal spacing g = 100 mm, see

Fig. 1(f). Nominal stress σN and nominal strain εN are defined in Eqs. (7) and

(8) where u =mean shortening of the specimen, D is the mean of the specimen

dimensions in the respective axis, obtained from four measurements.

σN = κF/D2, κ

1.0 cube

4/π cylinder(7)

εN =

u/D cube

u/g cylinder(8)

In Fig. 4 the nominal stress σN versus nominal strain εN diagrams for

uniaxial compression tests of 40 mm cubes, 150 mm cubes and 75x150 mm

cylinders are shown in terms of mean response curves and envelope. Speci-

men dimensions, peak stresses and elastic moduli are given by mean values

and coefficients of variation. Figs. 4(a-b) show nominal strain as derived from

on-specimen measurements while the strain values plotted in Figs. 4(c-d) are

derived from platen to platen measurements and include a compliance correc-

tion according to Eq. (6).

For many practical applications (e.g. the analysis of pre-damaged struc-

tural members in particular under cyclic loading) also the unloading-reloading

behavior in the softening regime is of interest. Fig. 5 provides insight into the

damage evolution of unconfined compression tests. Figs. 5(a-b) exemplarily

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


strain εN [10−3]

stress

σN

[MPa]

D =76.5 mm ±0.9%L =152.4 mm ±0.0%Ec =27735.5 MPa ±6.2%fc =46.6 MPa ±3.0%number of specimen: 9

0 1 2 30

10

20

30

40

50

60

strain εN [10−3]

stress

σN

[MPa]

D =76.1 mm ±0.3%L =152.4 mm ±0.0%Ec =34382.2 MPa ±3.9%fc =55.6 MPa ±3.7%number of specimen: 12

0 1 2 30

10

20

30

40

50

60

strain εN [-]

stress

σN

[MPa]

W =39.8 mm ±0.8%H =42.7 mm ±1.6%D =40.8 mm ±1.4%Ec =40509.0 MPa ±32.7%fc =59.0 MPa ±6.9%number of specimen: 5

0 0.005 0.01 0.015 0.020

10

20

30

40

50

60

70

strain εN [-]

stress

σN

[MPa]

W =151.7 mm ±0.4%H =152.3 mm ±0.1%D =152.8 mm ±0.2%Ec =38063.2 MPa ±7.6%fc =58.7 MPa ±5.6%number of specimen: 4

0 0.005 0.01 0.015 0.020

10

20

30

40

50

60

70

Fig. 4 Nominal stress σN versus nominal strain εN for uniaxial compression tests: a) cylin-der 75x150 mm at 31 days, (b) cylinder 75x150 mm at 400 days, (c) cubes 40x40 mm at470 days, and (d) cubes 150x150 mm at 470 days

show the nominal stress-strain curves of single cube specimens under hys-

teretic loading conditions while Figs. 5(c-d) present the evolution of relative

loading and unloading moduli E/E0 plotted against the relative stress level

at the point of unloading f/f0. E0 is the initial loading modulus and f0 is

the peak stress value. The loading and unloading slopes are obtained through

linear regression in the ranges 0.40f < σ < 0.80f and 0.05f < σ < 0.80f ,

respectively. The modulus decrease shows a high linear correlation with the

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


W =39.7 mm ±0.9%H =42.4 mm ±1.3%D =41.1 mm ±0.9%fc =55.3 MPa ±10.5%number of specimen: 9

strain εN [-]

stress

σN

[MPa]

0 0.01 0.02 0.03 0.040

10

20

30

40

50

60

70

W =151.8 mm ±0.6%H =152.4 mm ±0.2%D =152.3 mm ±0.5%fc =55.6 MPa ±4.3%number of specimen: 4

strain εN [-]

stress

σN

[MPa]

0 0.01 0.02 0.03 0.040

10

20

30

40

50

60

70

f/f0

E/E

0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1loadingunloading

f/f0

E/E

0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1loadingunloading

Fig. 5 Uniaxial compression tests with hysteretic loading-unloading cycles: Nominal stressσN versus nominal strain εN for selected curves of (a) 40 mm cubes at 470 days, and (b)150 mm cubes at 470 days; the relative change in modulus is given in (c) for the 40 mmcubes, and in (d) for the 150 mm cubes

decrease in residual strength. Also, the unloading modulus is systematically

lower than the reloading modulus.

3.5 Confined compression

In addition to uniaxial unconfined compression ideally also triaxial and hydro-

static test data should be available for calibration. Unfortunately, it was not

possible to perform this type of test within the scope of the presented investi-

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


gation. The closest alternative data source that could be obtained are passively

confined compression tests conducted on cored cylinders. High passive confine-

ment was achieved using thick-walled steel jackets, see Fig. 6(a). In total four

confined and four unconfined but otherwise identical specimens were tested.

The documented specimen diameter and height are D = 43.7 mm ±0.1% and

L = 38.4 mm ±2.4%, respectively. The steel jacket is 88.6 mm high, has an

inner diameter of 47.3 mm and a wall thickness of 14.15 mm. The plunger and

the bottom loading block provide a very tight fit with a diameter of 47 mm.

For the confined tests the cylinders were grouted into the steel jacket with

quickcrete, resulting in a capped height of L = 41.4 mm ±2.8%. The uncon-

fined partner specimens were not capped. In addition to force F and piston

stroke u also the circumferential expansion ∆p of the steel jacket as indicator

for the confinement was recorded.

Fig. 6(b) shows the nominal stress strain diagram for the unconfined tests

utilizing the definition as introduced in Eqs. (7) and (8), while Fig. 6(c) presents

the response of the unconfined companion specimens (without steel jacket)

with otherwise identical setup to maintain the compliance contribution of the

setup, see Fig. 6(d).

Nominal strain is derived from stroke measurements without correction

for the compliance of the test setup. Compared to other uniaxial compression

tests, the unconfined cored cylinders show a slight decrease in compressive

strength. The response under strong confinement is strongly biased by the

grouting material and friction, thus reducing the value of this data.

3.6 Flexural fracture by 3-point-bending

A major part of the presented investigation concerns the characterization of

flexural fracture. In particular, the size dependency of flexural strength and

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


Fig. 6 Compression test under strong passive confinement; (a) test setup for confined test,(b) response under confinement, (c) unconfined response of partner specimen, and (d) setupfor unconfined companion specimens

toughness are of interest. Further studied parameters include the relative notch

depth α = a/D, relative load eccentricity ξ = x/l, and the modulus reduc-

tion in the softening regime. In total 128 geometrically scaled beams of four

sizes with a size range of 1:12.5 were tested. In addition to un-notched spec-

imens, notch depths of α = 0.3, 0.15, 0.075 for all sizes and α = 0.025 for

the two larger sizes were investigated, see also [53]. For each size and notch

depth combination at least six specimens were tested, more for the smallest

two sizes due the larger inherent scatter. The specimen geometry is sketched

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


Table 4 Nominal geometry of 3-point-bending specimens

property A B C D[mm] [mm] [mm] [mm]

thickness, W 40.0 40.0 40.0 40.0height, D 500 215 92.8 40.0length, L 1200 517 223 96.0span, l 1088 469 202 87.0gauge length, g 162;218 94.5;137 60.0;25.4 25.4gauge length, g (α = 0.3) 25.4 25.4 12.7 12.7loading block width, w 60.0 26.0 11.0 5.2loading block height, h 40.0 20.0 10.0 5.0

in Fig. 1(a) and given in Table 4. Except notch width and specimen thick-

ness W all dimensions including the steel support blocks were geometrically

scaled. At an age of 96 days the notches were cut with a diamond coated

band saw. A lot of attention was placed on minimizing the time that the

specimens spent out of the 100% relative humidity room. The resulting notch

width is 1.8 mm. Microscopy revealed a notch-tip geometry that can be well

approximated by a half circle with diameter 2r = 1.8 mm, see Fig. 2(b). An

even better approximation is obtained with an ellipse of transverse diameter

2a = 1.8 mm and a conjugate diameter 2b = 1.3 mm. In addition to the ge-

ometrically scaled beams a total of 12 standard ASTM modulus of rupture

tests [55] were performed in stroke control. The average dimensions according

to Fig.1(a) are D = 152.6mm ± 0.4% W = 152.9mm ± 0.3% with a nominal

length of L = 558.8 mm and a span of l = 457.2 mm. The loading rate was

chosen in such a way that the notch tip would be strained at roughly the same

rate for all specimens, corresponding to a loading time of roughly 5 minutes

to peak. For stability reasons the control method was switched from stroke to

crack mouth opening at about 60% of the predicted peak.

Approximately 400 days after casting all 128 beams of the bending size

effect investigation were tested within a span of 11 days. The chosen stable

mode of control was CMOD for notched specimens and average tensile strain

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


for un-notched beams. Specimens of sizes C and D were loaded in a 90 kN

load frame whereas the larger specimens of sizes A and B were tested in the

1000 kN load frame. Fig. 7(a) gives a visual overview of the set of beams.

Before testing the initial dimensions of all specimens were recorded rigorously

and the steel loading blocks glued onto the top and bottom surfaces. Avail-

able sensor information included for all tests, in addition to machine force

and stroke, center-point displacement, obtained by averaging two LVDT mea-

surements against the load bed, and extensometer readings on the tension

side of the beam. The latter correspond to crack mouth opening displacement

(CMOD) measurements for the deeper notch depths where the elastic defor-

mation within the gauge length g given in Table 4 is negligible. Specimens with

shallower notches and especially un-notched specimens were instrumented with

sensors of larger gauge length g to ensure a crack localization within.

The specimen response is plotted in terms of nominal stress σN versus

nominal strain εN for the un-notched specimens of all four sizes in Fig. 7(b).

The corresponding plots of notched specimens with relative notch lengths of

α = 0.025, 0.075, 0.15, 0.3 are given in Figs. 7(c-f). For the purpose of this

investigation nominal stress σN for bending specimens is defined according

to beam theory by Eq. (9) where both D and W are obtained as mean value

of 3 and 5-9 measurements at the ligament, respectively. The nominal strain,

εN , definitions are based on the measured opening u of the extensometer with

gauge length g and are given in Eq. (10). For specimens with deep notch the

crack mouth opening can be fairly well approximated by the extensometer

reading at the surface of the beam which is proportional to the CMOD and

which is only slightly distorted by elastic deformations in the basically stress

free concrete next to the cut. The larger the specimen the closer the extensome-

ter feet can be set to the notch and the smaller this influence becomes. For

shorter notches (and especially small specimen) elastic deformations within

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


the gauge length gain in importance but remain small compared to the to-

tal extensometer reading u. Furthermore, these elastic deformations scale if

the gauge length scales with specimen size as approximately true in this in-

vestigation. Thus, they do not influence size effect investigations and can be

neglected in the definition of nominal strain for notched specimens. However,

for the sake of accuracy the gauge length (and consequently the contributions

of elastic deformations) should be modelled in numerical simulations if model

calibration or inverse analysis are to be performed.

For un-notched specimens an engineering strain definition is chosen. In this

case, since the gauge length is finite and not negligible compared to span length

l a correction factor β is required in order to account for the linear moment

distribution in between the extensometer feet with spacing g by converting

the average measured strain to peak strain in mid-span. This is especially

important since the gauge lengths did not scale with size for this set of tests.

σN =6F (1− ξ)ξl

W D2(9)

εN =

βu/g for α = 0

u/D for α > 0(10)

As expected, for increasing specimen size D, strength decreases and the

post-peak regime shows a transition from ductile to rather brittle behavior.

For all five geometrically similar beam sets the slope in the elastic regime

coincides. The traditional presentation of the size effect in flexural strength in

the form of a log(σN,peak) versus log(D) plot is omitted here for the sake of

brevity and can be found, together with an extensive analysis and discussion,

in Hoover et al. [53,60,66].

Concrete is a highly heterogenous material. As such, a significant scatter

in structural response due to the random distribution of strength is expected.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


The macroscopic strength of specimens without initial notch is influenced most

by the material heterogeneity which manifests itself in a wide-spread crack

localization on the tension side, see the photogrammetrically obtained crack

path distribution of all un-notched specimens in Fig. 8. This variability of

local strength is also the cause of statistical distribution of strength, which

is, for small material elements, Gaussian with a remote power law tail that

gives rise to Weibull distribution [67] when the structure is very large and

fails at fracture initiation [68,69]. In comparison, Figs. 9 to 12 show the crack

distributions of the notched beams with all cracks emanating from the notch

tip. The grey lines represent the crack path on each surface, while the solid

black lines denote the average crack path x(y) over the ligament depth y in a

x− y coordinate system, with origin at the crack tip or at the centerpoint on

the tension surface, respectively.

Data about the bending shear interaction are available in the form of ec-

centrically loaded un-notched beams of the second smallest size with depth

D = 93 mm. Fig. 13(a) shows nominal stress strain curves for load eccentric-

ities of 47 mm and 81 mm corresponding to ξ = x/l = 0.53, 0.20 tested at

an age of 466 days in comparison with the original centrically loaded beams.

Fig. 13(b) presents an example of an un-notched beam of the same size with

loading-unloading cycles in the softening regime. Figs. 13(c-d) finally show

the modulus reduction with softening damage of hysteretically loaded beams

of size D=93 mm. The analysis is based on the concept presented in section

3.4. All additional specimens were instrumented with an extensometer of gauge

length g = 44.4 mm.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


Fig. 7 Three point bending: (a) size comparison, (b) nominal stress-strain diagram forun-notched beams, (c) for beams with α = 2.5%, (d) α = 7.5%, (e) α = 15%, and (f)α = 30%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

Fig. 8 Crack pattern for un-notched beams of (a) size D = 40 mm, (b) size D = 93 mm,(c) size D = 215 mm, (d) size D = 500 mm

3.7 Brazilian Splitting tests

The so-called Brazilian splitting test represents another commonly used form

of indirect tensions tests in which the specimen is loaded in compression, lead-

ing to tensile stresses and ultimately tensile failure in the direction orthogo-

nal to the applied load. The ASTM standard C496 [70] recommends testing

cylinders radially under point loads. The specimens are supported and loaded

by wooden bearing strips of pre-determined dimensions. While this practice

improves the stability of the tests and avoids local failure due to stress singu-

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 9 Crack pattern for beams of notch-depth α= 0.30 of (a) size D = 40 mm, (b) sizeD = 93 mm, (c) size D = 215 mm, (d) size D = 500 mm

larities, the boundary conditions become ambiguous. Contrarily, the European

code EN12390-6 [71] demands direct loading of cylinders by loading blocks, or

prisms by loading cylinders. In both cases the curvature ratio is maintained,

thus ensuring well defined and equivalent boundary conditions. In the idealized

case of elastic response under a concentrated load (β = 0) the theoretically

maximum tensile stress is given by

σmax(β = 0) =2F

πWD(11)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


If the material were perfectly brittle this equation would give the tensile

strength ft. However, since concrete is a quasi-brittle material this value does

not coincide with the true tensile strength but is expected to be close [72]. It

is thus defined as splitting tensile strength fst.

In reality the contact area between support and specimen is always finite

with a relative bearing width β = w/D. Furthermore, standards recommend

the use of bearing strips (0.04 ≤ β ≤ 0.16) to distribute the loads and avoid

local crushing. This distribution of loading leads, for the same total load,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


to a more uniform stress state in the center of the specimen parallel to the

direction of loading and, in consequence, to a reduction in principal tensile

stress in the perpendicular direction. This results in a tensile strength increase

for increasing bearing strip width and is captured by the correction term κ(β)

that was deduced by Tang for cylindrical specimens [73]. For square prismatic

specimens Rocco et al. [74,72] obtained an empirical equation valid for β <

0.20.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ξ = x/D[−]

η=

y/D[−

]

−0.2 −0.1 0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 12 Crack pattern for beams of notch-depth α= 0.025 of (a) size D = 215 mm, (d) sizeD = 500 mm

σmax(β) = κ(β)2F

πWD, κ(β) =

(1− β2)3/2 for cylinder

(1− β2)5/3 − 0.0115 for prisms(12)

Within this investigation prismatic specimens according to Fig. 1(b) of

five sizes with constant thickness W = 40 mm were tested. In addition to

the sizes D = 40, 93, 215, 500 of the 3-point-bending tests, specimens of size

D = 30 were tested, increasing the size range to 1:16.7. Two versions of bear-

ings strips with a relative bearing strip width β ≈ 0.08 were compared - oak

wood and steel. While an average of 6 specimens were tested for each size

and wood bearing strips, only 1-2 specimens with steel supports were tested

as reference. Unlike the bearing strip width the CMOD gauge length could

not be geometrically scaled due to the limited sensor availability. The nomi-

nal specimen dimensions, bearing strip width and gauge lengths are given in

Table 5. In the case of oak wood bearing strips the actual average dimensions

before and after the tests are given as well.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


strain εN [10−3]

stress

σN

[MPa]

3ptbending, unnotched (SE)

ξ = 0.50 (e = 0mm)

ξ = 0.27 (e = 47mm)

ξ = 0.10 (e = 81mm)

0 1 2 3 40

1

2

3

4

5

6

7

8

9

strain εN [10−3]

stress

σN

[MPa]

0 1 2 30

1

2

3

4

5

6

7

8

9

f/f0

E/E

0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1loadingunloading

f/f0

E/E

0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1loadingunloading

Fig. 13 Three point bending of specimens with D=93 mm: (a) nominal stress strain plotof un-notched beams dependent on eccentricity; (b) example of hysteretically loaded un-notched beam; relative modulus versus relative stress of (c) un-notched beams, and (d)beams with α = 0.3

Table 5 Geometry of Brazilian splitting test specimens

property A B C D E[mm] [mm] [mm] [mm] [mm]

thickness, W 40.0 40.0 40.0 40.0 40.0height, width, D 500 215 92.8 40.0 30.0gauge length, g 34.0 34.0 26.0 16.0 8.0steel bearing strip width, w 60.0 26.0 11.0 5.2 5.2steel bearing strip height, h 40.0 20.0 10.0 5.0 5.0wood, initial w 37.6 16.7 7.6 2.7 2.6wood, initial h 18.4 8.0 3.5 2.7 1.7wood, final w 46.3 20.1 9.4 4.8 3.5wood, final h 9.4 4.2 1.9 1.2 1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


The specimen response is plotted in terms of nominal stress σN versus

nominal strain εN for all five sizes and wooden bearing strips in Fig. 14(a).

For the purpose of this investigation nominal stress σN is defined as maximum

tensile stress σmax(β) according to Rocco et al. [74,72] and Eq. (12). Nominal

strain is defined as engineering strain utilizing the extensometer opening u and

the gauge length g:

εN = u/g (13)

As expected, the nominal stress σN decreases with sizeD and the post-peak

response shows a transition from highly ductile to a rather brittle behavior.

The nominal strength values of all sizes for both bearing strip versions are

reported in Table 6 in terms of mean value and coefficient of variation. In

all cases the final, not the initial, width of the bearing strip was used in the

determination of nominal stresses. The observed normalized crack patterns for

all specimens with wooden bearing strips are given in Figs. 14(b-f) according

to size. It must be observed that the origin of the coordinate system was chosen

in the center of the prism where the highest tensile stresses are to be expected,

lacking a better reference point that can be identified on all specimens. Thus,

all main cracks (bold lines) seemingly emanate from this point. The secondary

cracks that likely formed after the specimen already failed have been recorded

relative to the main crack and are presented as dashed lines.

For the analysis only specimens with a stable opening after crack initiation

were considered, thus plotted in Fig. 14(a), and included in the statistics of

Table 6. Contrary to earlier analyses e.g. by Bazant et al. [75] a size-effect

is observed but has to be characterized as very mild. Direct loading without

wooden bearing strips resulted in a generally lower tensile strength, in spite

of the compensation for the bearing strip width. The largest specimen size

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


Table 6 Splitting tensile strength according to size

label specimen size wood steel number of specimens[mm] [MPa] [MPa] (wood/steel)

A 500 4.5±4.5% 5.2 (5/2)B 215 5.1±6.0% 4.9 (5/1)C 93 5.1±2.5% 4.9 (7/-)D 40 5.2±6.3% 5.0 (9/2)E 30 5.1±14.9% 4.5 (7/2)

showed an increase in splitting strength. However, due to the low number of

specimens, at best qualitative conclusions can be drawn.

3.8 Torsion

Pure torsion test data were obtained for three cross-sections of widthW =40 mm

and depth H = 40, 60, 80 mm. The specimens were loaded by two opposing

moment couples with eccentricity 2e = 20mm as sketched in Fig. 1(e). Free

rotation of all four loading and support points was guaranteed by ball bear-

ings. The span length l was chosen to be 160 mm for the square cross-section

and 240 mm otherwise.

The average torsional capacity of the smallest cross-section is Tmax =

52.7 ± 0.3% Nm. An increase of the cross-sectional depth to H = 60 mm

and H = 80 mm resulted in an increase of load carrying capacity to Tmax =

84.3±3.1% Nm and Tmax = 110±0.2% Nm, respectively. The cracking angle on

the initiation face of specimens with rectangular cross–section was determined

to be 54.7◦ for the 80x40 and 49.3◦ for the 60x40 cross-section. The cracking

angle on the opposite faces was measured to be 37.0◦ and 30.0◦, respectively.

Nominal stress may be defined according to the elastic solution for torsion

of rectangular cross sections as shear stress τ , based on the mean values of

eccentricity e, width W , and depth D, and the torsional moment of inertia

JT . The influence of the aspect ratio D/W is captured by the dimensionless

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


Fig. 14 Brazilian splitting tests; (a) mean nominal stress strain curves and envelopes ofspecimens with D = 500, 215, 93, 40, 30 mm; crack pattern for (b) D = 30mm, (c) D =40mm, (d) D = 93mm, (e) D = 215mm, and (f) D = 500mm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


correction factor β = β(D/W ) (β = 0.208 for D/W = 1, β = 0.231 for

D/W = 1.5, and β = 0.246 for D/W = 2).

σN = τ =MT r

JT=

F e

βW 2D(14)

The corresponding nominal strain is defined based on the rate of the angle

of twist ϑ′ by

εN = ϑ′r =δ

e · lD

2(15)

where δ = measured load point displacement.

3.9 Direct tension tests

Single notch direct tension tests on approximately 950 day old concrete prisms,

cut from undamaged pieces of the original size effect investigation, were per-

formed. The specimen dimensions are given in Table 7. In order to ensure

localization of the crack in the center cross-section and thus be able to control

the test by crack mouth opening a notch with a relative depth α = a/D = 0.25

was cut, see Fig. 15(a). The boundary conditions are characterized by fully

clamped support on one side and pin support on the other end as sketched in

Fig. 1(g).

The nominal stress is defined according to the elastic solution of un-notched

specimens according to Eq. (16) while nominal strain is defined as engineering

strain based on the measured extensometer opening u. The resulting nominal

stress-strain diagram is given in Fig. 15 with one removed outlier.

σN =F

WD(16)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


Fig. 15 Single notch tension tests; (a) image of setup; nominal stress σN , nominal strainεN diagram.

Table 7 Specimen geometry

property mean[mm]

thickness, W 25.6± 1.1%height, D 26.4± 2.5%length, L 127.4± 0.4%free length, l 74.5± 3.5%notch depth, a 6.4± 7.8%gauge length, g 18.9± 1.1%

εN = u/g (17)

4 Conclusion and Outlook

A comprehensive data-set consisting of 152 notched and un-notched beams, 32

cylinders, 28 cubes, 46 Brazilian splitting prisms, 6 single notch tension tests,

and 11 torsion tests has been presented. Post-peak data and failure mode in

terms of crack patterns are available for almost every specimen tested. With

a size range of 1:12.5 for the bending investigation and 1:16.7 for the split-

ting tests, this data-set contains some of the largest size-effect investigations on

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


plain concrete, and thus represents an ideal data source for model development

and calibration which can be found at ’http://www.baunat.boku.ac.at/cd-

labor/downloads/versuchsdaten’.

Acknowledgements The financial support by the Austrian Federal Ministry of Economy,

Family and Youth and the National Foundation for Research, Technology and Development

for part of the analysis is gratefully acknowledged, as is the financial support from the

U.S. Department of Transportation, provided through Grant 20778 from the Infrastructure

Technology Institute of Northwestern University, for the initial size effect investigation. The

work of G. Cusatis was supported under NSF grant No 0928448.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


List of Figures

1 Specimen geometry: (a) three point bending tests, geometrically

scaled in four sizes, (b) Brazilian splitting tests, geometrically

scaled in five sizes, (c) ASTM modulus of rupture tests [55], (d)

unconfined compression of cubes of two sizes, (e) torsion tests,

(f) unconfined cylinder tests of two sizes, and (g) single notched

tension tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Stability of fracture tests: (a) observed, inelastic, corrected, and

critical response for 93 mm beams and α = 0.15, (b) observed

stable load vs. opening curves for 93 mm beams and α = 0.15 . 16

3 3-point bending: (a) normalized load-CMOD curves and average

response curve by the example of 3-point bending tests with

D = 93mm and α = 0.15, (b) microscopic image of cut notch tip 19

4 Nominal stress σN versus nominal strain εN for uniaxial com-

pression tests: a) cylinder 75x150 mm at 31 days, (b) cylinder

75x150 mm at 400 days, (c) cubes 40x40 mm at 470 days, and

(d) cubes 150x150 mm at 470 days . . . . . . . . . . . . . . . . 22

5 Uniaxial compression tests with hysteretic loading-unloading

cycles: Nominal stress σN versus nominal strain εN for selected

curves of (a) 40 mm cubes at 470 days, and (b) 150 mm cubes

at 470 days; the relative change in modulus is given in (c) for

the 40 mm cubes, and in (d) for the 150 mm cubes . . . . . . . 23

6 Compression test under strong passive confinement; (a) test

setup for confined test, (b) response under confinement, (c) un-

confined response of partner specimen, and (d) setup for uncon-

fined companion specimens . . . . . . . . . . . . . . . . . . . . 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


7 Three point bending: (a) size comparison, (b) nominal stress-

strain diagram for un-notched beams, (c) for beams with α =

2.5%, (d) α = 7.5%, (e) α = 15%, and (f) α = 30% . . . . . . . 30

8 Crack pattern for un-notched beams of (a) size D = 40 mm,

(b) size D = 93 mm, (c) size D = 215 mm, (d) size D = 500 mm 31

9 Crack pattern for beams of notch-depth α= 0.30 of (a) size

D = 40 mm, (b) size D = 93 mm, (c) size D = 215 mm, (d)

size D = 500 mm . . . . . . . . . . . . . . . . . . . . . . . . . . 32



size D = 500 mm . . . . . . . . . . . . . . . . . . . . . . . . . . 33



size D = 500 mm . . . . . . . . . . . . . . . . . . . . . . . . . . 34


D = 215 mm, (d) size D = 500 mm . . . . . . . . . . . . . . . . 35

13 Three point bending of specimens with D=93 mm: (a) nomi-

nal stress strain plot of un-notched beams dependent on eccen-

tricity; (b) example of hysteretically loaded un-notched beam;

relative modulus versus relative stress of (c) un-notched beams,

and (d) beams with α = 0.3 . . . . . . . . . . . . . . . . . . . . 36

14 Brazilian splitting tests; (a) mean nominal stress strain curves

and envelopes of specimens with D = 500, 215, 93, 40, 30 mm;

crack pattern for (b) D = 30mm, (c) D = 40mm, (d) D =

93mm, (e) D = 215mm, and (f) D = 500mm . . . . . . . . . . 39

15 Single notch tension tests; (a) image of setup; nominal stress

σN , nominal strain εN diagram. . . . . . . . . . . . . . . . . . . 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


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